I'm looking for topics in measure theory that are related to set theory (descriptive set theory or otherwise). It seems that there's a lot of connection between topology and set theory and I'm wondering if the same is true for measure theory.
Edit: To be more specific, can one give some examples of areas where these subjects are related? For example, the only area I'm aware of is cardinal characteristics of the continuum where people study at which level, i.e. ordinal, additivity of certain measures on the reals fail.
It is important. For example look at the work of Alexander Kechris from Caltech. Also see the work of Furstenberg and Pestov.